A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Parsimonious Path Openings and Closings

International audiencePath openings and closings are morphological tools used to preserve long, thin and tortuous structures in gray level images. They explore all paths from a defined class, and filter them with a length criterion. However, most paths are redundant, making the process generally slow. Parsimonious path openings and closings are introduced in this paper to solve this problem. These operators only consider a subset of the paths considered by classical path openings, thus achieving a substantial speed-up, while obtaining similar results. Moreover, a recently introduced one dimensional (1-D) opening algorithm is applied along each selected path. Its complexity is linear with respect to the number of pixels, independent of the size of the opening. Furthermore, it is fast for any input data accuracy (integer or floating point) and works in stream. Parsimonious path openings are also extended to incomplete paths, i.e. paths containing gaps. Noise-corrupted paths can thus be processed with the same approach and complexity. These parsimonious operators achieve a several orders of magnitude speed-up. Examples are shown for incomplete path openings, where computing times are brought from minutes to tens of milliseconds, while obtaining similar results

Skeletonization methods for image and volume inpainting

Image and shape restoration techniques are increasingly important in computer graphics. Many types of restoration techniques have been proposed in the 2D image-processing and according to our knowledge only one to volumetric data. Well-known examples of such techniques include digital inpainting, denoising, and morphological gap filling. However efficient and effective, such methods have several limitations with respect to the shape, size, distribution, and nature of the defects they can find and eliminate. We start by studying the use of 2D skeletons for the restoration of two-dimensional images. To this end, we show that skeletons are useful and efficient for volumetric data reconstruction. To explore our hypothesis in the 3D case, we first overview the existing state-of-the-art in 3D skeletonization methods, and conclude that no such method provides us with the features required by efficient and effective practical usage. We next propose a novel method for 3D skeletonization, and show how it complies with our desired quality requirements, which makes it thereby suitable for volumetric data reconstruction context. The joint results of our study show that skeletons are indeed effective tools to design a variety of shape restoration methods. Separately, our results show that suitable algorithms and implementations can be conceived to yield high end-to-end performance and quality of skeleton-based restoration methods. Finally, our practical applications can generate competitive results when compared to application areas such as digital hair removal and wire artifact removal

Efficient robust d-dimensional path operators

International audiencePath openings and closings are efficient morphological operators that use flexible oriented paths as structuring elements. They are employed in a similar way to operators with rotated line segments as structuring elements, but are more effective at detecting linear structures that are not necessarily locally perfectly straight. While their theory has always allowed paths in arbitrary dimensions, de facto implementations were only proposed in 2D. Recently, a new implementation was proposed enabling the computation of efficient -dimensional path operators. However this implementation is limited in the sense that it is not robust to noise. Indeed, in practical applications, for path operators to be effective, structuring elements must be sufficiently long so that they correspond to the length of the desired features to be detected. Yet, path operators are increasingly sensitive to noise as their length parameter increases. To cope with this limitation, we propose an efficient -dimensional algorithm, the Robust Path Operator, which uses a larger and more flexible family of flexible structuring elements. Given an arbitrary length parameter G, path propagation is allowed if disconnections between two pixels belonging to a path is less or equal to G and so, render it independent of . This simple assumption leads to constant memory bookkeeping and results in a low complexity